Solution:

By symmetry, $AG=HC=HJ=1$, and triangles $AFG$ and $BGH$ are congruent, so $FG=GH$. Let $FG=b$, and let $CD=d$.

The similar triangles in the pentagon give $$\frac1b=1+b$$ and $$\frac{1+b}{1}=d.$$ The first equation is $b^2+b-1=0$, so $b=\frac{-1+\sqrt{5}}{2}$. Then $d=1+b=\frac{1+\sqrt{5}}{2}$.

Therefore $$FG+JH+CD=b+1+d=1+\sqrt{5}.$$

Thus, the correct answer is D.